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arXiv:2609.40095

Classical Shadows with Selective GHZ Measurements

Nicolas Faroß, Marc Wanner, Pingal Pratyush Nath, Akshay Gaikwad, Anton Frisk Kockum, Devdatt Dubhashi

quant-phSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

Classical shadows allow the prediction of many quantum-state properties from measurement data, but their efficiency strongly depends on the underlying measurement ensemble. We introduce selective GHZ measurement (SGM) shadows, which perform Greenberger-Horne-Zeilinger (GHZ)-basis measurements on randomly selected subsets of qubits and computational-basis measurements on their complement. The corresponding quantum circuits are practically feasible, with their number of CNOT gates scaling linearly with the number of qubits. This is in stark contrast to global Clifford circuits, which are generally difficult to implement. Varying the subset distribution allows tuning the SGM shadow protocol for different classes of observables with prescribed X/Y support, allowing us to outperform local Pauli as well as global Clifford shadows in certain scenarios. We determine the exact shadow channel and its inverse for arbitrary subset distributions and prove an explicit bound on the shadow norm using methods from discrete Fourier analysis and Markov semigroup theory. For natural choices of subset distributions, this gives a polynomial scaling for observables with fixed X/Y-locality. Numerical simulations of up to 50 qubits support the predicted improvements for nonlocal observables and long-range systems, and demonstrate the practical benefits of our protocol's generality by showing how the subset distributions can be tailored to settings such as fermionic modes under the Jordan-Wigner transformation.

Comments
53 pages, 4 figures

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Classical shadows allow the prediction of many quantum-state properties from measurement data, but their efficiency strongly depends on the underlying measurement ensemble. We introduce selective GHZ measurement (SGM) shadows, which perform Greenberger-Horne-Zeilinger (GHZ)-basis measurements on randomly selected subsets of qubits and computational-basis measurements on their complement. The corresponding quantum circuits are practically feasible, with their number of CNOT gates scaling linearly with the number of qubits. This is in stark contrast to global Clifford circuits, which are generally difficult to implement. Varying the subset distribution allows tuning the SGM shadow protocol for different classes of observables with prescribed X/Y support, allowing us to outperform local Pauli as well as global Clifford shadows in certain scenarios. We determine the exact shadow channel and its inverse for arbitrary subset distributions and prove an explicit bound on the shadow norm using methods from discrete Fourier analysis and Markov semigroup theory. For natural choices of subset distributions, this gives a polynomial scaling for observables with fixed X/Y-locality. Numerical simulations of up to 50 qubits support the predicted improvements for nonlocal observables and long-range systems, and demonstrate the practical benefits of our protocol's generality by showing how the subset distributions can be tailored to settings such as fermionic modes under the Jordan-Wigner transformation.
Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U. In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ``real'' classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.
Abstract Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U . In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k -local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in- k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.
We provide practical and powerful schemes for learning many properties of an unknown n-qubit quantum state using a sparing number of copies of the state. Specifically, we present a depth-modulated randomized measurement scheme that interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. These can be seen within our scheme as the special cases of zero and infinite depth, respectively. We focus on the regime where depth scales logarithmically in n and provide evidence that this retains the desirable properties of both extremal schemes whilst, in contrast to the random Clifford scheme, also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We consider observables that can be written as a linear combination of poly(n) Paulis and observables that can be written as a low bond dimension matrix product operator. For the former class of observables both tasks are solved efficiently in n. For the latter class, we do not guarantee efficiency but present a method that works in practice; by variationally computing a heralded approximate inverses of a tensor network that can then be used for efficiently executing both these tasks.
Classical shadows use a single measurement snapshot to estimate many observables. These estimates are jointly distributed, and their covariance is not determined by their expectation values or individual variances. It also differs from physical correlations between observables because it depends on the measurement and reconstruction protocol. We study the covariance matrix for a fixed set of shadow estimates and its spectrum. For general shadow protocols, we give a finite-sample operator-norm bound for the empirical covariance computed using the sample mean. This bound also controls the eigenvalues and isolated spectral subspaces of the covariance matrix. For local product shadow protocols with fixed local dimension, observables supported on a bounded number of sites lead to bounds that do not depend on the total system size, provided that the number of selected observables and the relevant local reconstruction factors remain bounded. For biased local Pauli shadows, we obtain explicit finite-sample bounds and an exact covariance formula determined by Pauli compatibility and local basis probabilities. A comparison with global Clifford shadows shows that dimension-independent covariance estimation depends on the measurement protocol and does not follow from the general theory alone.
Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.
We study classical shadows protocols based on randomized measurements in n -qubit entangled bases, generalizing the random Pauli measurement protocol ( n=1 ). We show that entangled measurements ( n≥2 ) enable nontrivial and potentially advantageous trade-offs in the sample complexity of learning Pauli expectation values. This is sharply illustrated by shadows based on two-qubit Bell measurements: the scaling of sample complexity with Pauli weight k improves quadratically (from ∼3k down to ∼3k/2 ) for many operators, while others become impossible to learn. Tuning the amount of entanglement in the measurement bases defines a family of protocols that interpolate between Pauli and Bell shadows, retaining some of the benefits of both. For large n , we show that randomized measurements in n -qubit GHZ bases further improve the best scaling to ∼(3/2)k , albeit on an increasingly restricted set of operators. Despite their simplicity and lower hardware requirements, these protocols can match or outperform recently-introduced "shallow shadows" in some practically-relevant Pauli estimation tasks.